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1: 1.4 Calculus of One Variable
If f ( x 1 ) f ( x 2 ) for every pair x 1 , x 2 in an interval I such that x 1 < x 2 , then f ( x ) is nondecreasing on I . … If f ( x ) is continuous on an interval I save for a finite number of simple discontinuities, then f ( x ) is piecewise (or sectionally) continuous on I . … If f ( n ) exists and is continuous on an interval I , then we write f C n ( I ) . … For α ( x ) nondecreasing on the closure I of an interval ( a , b ) , the measure d α is absolutely continuous if α ( x ) is continuous and there exists a weight function w ( x ) 0 , Riemann (or Lebesgue) integrable on finite subintervals of I , such that …
2: Bibliography Z
  • A. Zhedanov (1998) On some classes of polynomials orthogonal on arcs of the unit circle connected with symmetric orthogonal polynomials on an interval. J. Approx. Theory 94 (1), pp. 73–106.
  • 3: 26.2 Basic Definitions
    Unless otherwise specified, it consists of horizontal segments corresponding to the vector ( 1 , 0 ) and vertical segments corresponding to the vector ( 0 , 1 ) . For an example see Figure 26.9.2. … A partition of a set S is an unordered collection of pairwise disjoint nonempty sets whose union is S . … A partition of a nonnegative integer n is an unordered collection of positive integers whose sum is n . As an example, { 1 , 1 , 1 , 2 , 4 , 4 } is a partition of 13. …
    4: 28.17 Stability as x ±
    However, if ν 0 , then ( a , q ) always comprises an unstable pair. …
    5: 18.40 Methods of Computation
    See accompanying text
    Figure 18.40.1: Histogram approximations to the Repulsive Coulomb–Pollaczek, RCP, weight function integrated over [ 1 , x ) , see Figure 18.39.2 for an exact result, for Z = + 1 , shown for N = 12 and N = 120 . Magnify
    Here x ( t , N ) is an interpolation of the abscissas x i , N , i = 1 , 2 , , N , that is, x ( i , N ) = x i , N , allowing differentiation by i . … This is a challenging case as the desired w RCP ( x ) on [ 1 , 1 ] has an essential singularity at x = 1 . …
    6: Bibliography K
  • R. B. Kearfott (1996) Algorithm 763: INTERVAL_ARITHMETIC: A Fortran 90 module for an interval data type. ACM Trans. Math. Software 22 (4), pp. 385–392.
  • 7: 10.23 Sums
    provided that f ( t ) is of bounded variation (§1.4(v)) on an interval [ a , b ] with 0 < a < x < b < 1 . …
    8: 1.6 Vectors and Vector-Valued Functions
    A path is defined by 𝐜 ( t ) = ( x ( t ) , y ( t ) , z ( t ) ) , with t ranging over an interval and x ( t ) , y ( t ) , z ( t ) differentiable. … with ( u , v ) D , an open set in the plane. …
    9: 2.3 Integrals of a Real Variable
    k ( ) and λ are positive constants, α is a variable parameter in an interval α 1 α α 2 with α 1 0 and 0 < α 2 k , and x is a large positive parameter. …
    10: 18.1 Notation
    x , y , t real variables.
    w ( x ) weight function ( 0 ) on an open interval ( a , b ) .